#72254669

order	3520
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₅ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₁₁×#41 ⋊ C₂×C₅
Hall	#10623453 ⋊ C₅
Hall	C₁₁⋊C₅ ⋊ #585
Sylow	C₁₁ ⋊ C₅×#585

Subgroups

	subgroup	G/H
center	C₄	C₂³×AGL(1,𝔽₁₁)
commutator, socle, cosocle	C₂×C₁₁	C₂⁵×C₅
Frattini	C₂	C₂⁴×AGL(1,𝔽₁₁)
Fitting	C₂×C₁₁×#41	C₂×C₅
hypercenter	C₂×#41	AGL(1,𝔽₁₁)
hypocenter	C₁₁	C₅×#585
LCS	G ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₄ ◁ C₂×#41

1620 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 176

Polycyclic presentation:

2^5,5,2,11|a^2=g,e^2=g,c^a=cg,d^a=dg,d^b=dg,e^b=eg,h^b=h^10,h^f=h^4

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#585×AGL(1,𝔽₁₁), #72269561, #72269564, …
maximal normal subgroup of	#585×AGL(1,𝔽₁₁), #72269561, #72269564, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 5, 2, 11]);;
gap> SetPower(coll, 1, F.7);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 8, 2, F.8^10);;
gap> SetConjugate(coll, 8, 6, F.8^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);